A Multiple GNSS-based Orbit Determination Algorithm for ...

International Global Navigation Satellite Systems Society IGNSS Symposium 2009 Holiday Inn Surfers Paradise, Qld, Australia 1 – 3 December 2009

A Multiple GNSS-based Orbit Determination Algorithm for Geostationary Satellites Li Qiao School of Surveying & Spatial Information Systems, University of New South Wales, Sydney, Australia +61 2 9385 4174 & +61 2 9313 7493 [email protected]

Samsung Lim School of Surveying & Spatial Information Systems, University of New South Wales, Sydney, Australia +61 2 9385 4505 & +61 2 9313 7493 [email protected]

Chris Rizos School of Surveying & Spatial Information Systems, University of New South Wales, Sydney, Australia +61 2 9385 4205 & +61 2 9313 7493 [email protected]

Jianye Liu School of Automation Engineering, Nanjing University of Aeronautics and Astronautics, China +86 25 84892300 & +86 25 84895889 [email protected]

ABSTRACT The application of Global Positioning System (GPS) technology to the Geostationary Earth Orbit (GEO) determination has been constrained by the poor satellite visibility and weak signal power. This situation is expected to improve when multi-constellation Global Navigation Satellite Systems (GNSS) are available in the future. This paper aims to investigate a navigation algorithm to determine the GEO state vector in real time, using multi-constellation GNSS measurements. The navigation algorithm is based on a Kalman filter where the estimated state includes position and velocity corrections to the nominal reference trajectory and clock biases. The simulation results are presented in the paper. It is concluded that this algorithm meets the requirement for autonomous GEO satellite navigation.

KEYWORDS: Autonomous navigation, orbit determination, extended Kalman filter, spaceborne GPS. 1. INTRODUCTION Advances in autonomous satellite navigation technologies are essential in order to minimise the cost of operating satellites missions. The use of Global Navigation Satellite System (GNSS) signals is one of the most promising autonomous satellite navigation methods, being able to generate high accuracy ephemerides of geostationary satellites when equipped with a GNSS receiver.

Research on space-based applications of GNSS has been growing rapidly ever since the feasibility of using a spaceborne GPS receiver to autonomously determine a spacecraft orbit in real-time was demonstrated (Melbourne et al. 1994; Martin-Mur et al. 1995). Over the last decade satellite navigation using spaceborne GPS receivers has become a common technique for determining the ephemerides of Low Earth Orbit (LEO) satellites. Numerous spaceborne GPS receivers have been developed around the world, especially in the U.S. and in the European Union (Munjal et al. 1992; Bauer et al. 1998; Ebinuma & Unwin 2007; Krauss et al. 2008). After its successful use on LEO satellite applications there is a widespread interest in expanding the use of GPS to on-orbit navigation of satellites in highly eccentric or geostationary orbits. By several space experiments, the use of GPS/GNSS signals has been explored for different types of orbits, from LEO to Highly Elliptical Orbit (HEO) and Geostationary Satellite Orbit (GEO) (Eissfeller et al. 1996; Kronman 2000). However, there are two main, and closely connected, problems with GEO navigation using GNSS (Vasilyev 1999). Firstly, a GNSS signal received at the GEO satellite has very weak signal power. The second problem is the poor geometry of the GEO-GNSS satellites, as illustrated in Figure 1. This implies that the difference in the reception of GNSS signals by GEO satellite-borne receivers in comparison to LEO satellites lies in the geometric distance between a GNSS satellite and the navigating satellite vehicle (NSV) at high altitudes, where the GNSS satellites transmit signals towards the Earth. The GNSS signals attenuate significantly when received outside of the main lobe i.e. outside of 21.3 degrees from the line between the center of the Earth and the satellite (see Figure 1). Satellites at GEO orbits are located above the altitudes of the GNSS satellites, and therefore their reception of the signals is only possible from GNSS satellites on the other side of the Earth.

GEO Navigation Sat. Orbit NAV Sat 21.3°

Main lobe GEO satellite

Figure 1: Geometry of earth, GNSS satellites and GEO satellite. This GEO geometric situation will improve when multi-constellation GNSSs are fully operational in the future. This is because it is expected that there will be more than 100 GNSS satellites in orbit for positioning, in particular the satellites of the four GNSSs developed by U.S. (GPS), Russia (GLONASS), Europe Union (GALILEO) and China (COMPASS). This

paper assumes a multi-GNSS constellation consisting of 31 GPS satellites, 24 GLONASS satellites, 30 GALILEO satellites and 27 COMPASS satellites. Moreover, a multi-GNSS spaceborne receiver is feasible because of the planned interoperability of the different GNSSs, at least for some of the signals/frequencies. Hence the next generation GNSS receiver will likely be capable of processing not only GPS signals but also GLONASS, GALILEO and COMPASS signals (Krauss et al. 2008). It is assumed in this paper that a multi-GNSS receiver receives signals from all the systems and makes use of the pseudo-range measurements. This paper presents a navigation algorithm to determine the GEO state vector using multiconstellation GNSS pseudo-range measurements. A conventional approach for solving the GEO navigation problem is to take into account a satellite motion model and to perform the differential orbit correction and clock model parameter estimation. Experimental results are presented in this paper for the performance analysis. An improved navigation of the GEO satellite can be achieved using a dynamical model and sequential updates by a Kalman filter (Kronman 2000; Mittnacht & Fichter 2000; Mehlen & Laurichesse 2001). 2 ORBIT DETERMINATION STRATEGY 2.1 Measurement Models GNSS pseudo-range measurements are related to the satellite centre as the receiver location is assumed to be with respect to the satellite centre of mass in the satellite body-fixed frame. The measurement equation at each epoch is: ρ j = R j + δ tu + v ρ j

(1)

where j

ρj Rj

vρ j δ tu

GNSS satellite index pseudo-range from the GEO satellite to the navigation satellite

Sj

j geometrical distance from the GEO satellite to the navigation satellite S

random measurement noise range error equivalent to the receiver clock offset with respect to GNSS time frame

When n satellites are available, the filter measure model can be expressed as: Z = h( X ) + V

where

Z = [ ρ1

h( X ) = [ r1 + δ tu

ρ2 r2 + δ tu

ρn ]

T

rn

(2)

is the measurement vector, composed of measured pseudo-ranges ρ , + δ t ] is a function defining the relationship between the measurements j

T

u

=⎡

⎤

T

vρ ⎦ and the state vector, and V ⎣vρ vρ is the receiver noise. All the measurement perturbations not included in the model are considered as Gaussian errors with E[V ] = 0 and E[VV T ] = R

and

1

2

ri = ( x − xNSi ) 2 + ( y − y NSi ) 2 + ( z − z NSi ) 2

n

is the true distance between satellite and navigation

satellite. 2.2 Dynamic Models For the deterministic dynamic model for a GEO satellite, the satellite state dynamics are established for the state equation of the system with respect to the J2000.0 Earth Centered Inertial (ECI) coordinate system. The X and Z axes point toward the mean vernal equinox and

mean rotation axes of the Earth at January 1, 2000, at 12:00:00.00 UTC. J2000.0 = 2000 January 1.5 = JD 2451545.0 TDT (Terrestrial Dynamical Time). Assuming that the position vector is as:

r = [x

y

z]

T

and the velocity vector is

v = [vx , v y , vz ]T

, the orbit state vector can be written

T ⎡r ⎤ Xorbit = ⎢ ⎥ = ⎡⎣ x, y, z , vx , v y , vz ⎤⎦ ⎣v⎦

(3)

In a general form, the dynamical equation of the satellite state can be represented as: Xorbit = f orbit ( X, t ) + Worbit (t )

(4) where f orbit is the function that describes the satellite state dynamics, and Worbit (t ) is the process noise of the state equation. Ignoring the process noise Worbit (t ) , the first derivative of the state vector X therefore is: ⎡r ⎤ ⎡ v ⎤ Xorbit = f orbit ( Xorbit , t ) = ⎢ ⎥ = ⎢ ⎥ ⎣ v ⎦ ⎣a ⎦

(5)

where a is the acceleration of the satellite. The main components of the satellite acceleration a in the dynamical equation depend on the altitude of the satellite orbit. For a GEO satellite whose orbit altitude is about 36,000km, the acceleration will be perturbed by the two-body acceleration effect, the non-spherical earth gravitational potential effect, and the luni-solar gravitational effect. Thus, the total acceleration of a GEO satellite can be written as: a = a Earth + a Sun + a Moon + a w (6) where a Sun and a Moon are the third-body effects of the sun and moon respectively, a Earth includes the two-body effect and the non-spherical gravitational potential effect from the earth, and a w represents the high-order terms that are considered to be noise of the system model. Other perturbations (such as atmospheric drag, solar radiation, earth tidal, etc.) are negligible compared to the remaining effects in a w . The main zonal harmonics are J 2 , J 3 and J 4 in the non-spherical perturbation. The nonspherical gravitational potential a Earth can be represented in the ECI coordinate system as (Vallado & McClain 2001):

a Earth

2 3 4 ⎡ μ x⎡ z2 ⎞ z2 ⎞ z z2 z 4 ⎞⎤ ⎛R ⎞ 3⎛ ⎛R ⎞ 5⎛ ⎛R ⎞ 5⎛ ⎢ − e3 ⎢1 + J 2 ⎜ e ⎟ ⎜ 1 − 5 2 ⎟ + J 3 ⎜ e ⎟ ⎜ 3 − 7 2 ⎟ − J 4 ⎜ e ⎟ ⎜ 3 − 42 2 + 63 4 ⎟ ⎥ r ⎠ r ⎠r r r ⎠ ⎦⎥ ⎝ r ⎠ 2⎝ ⎝ r ⎠ 2⎝ ⎝ r ⎠ 8⎝ ⎢ r ⎢⎣ ⎢ 2 3 4 ⎢ μ y⎡ z2 z 4 ⎞⎤ z2 ⎞ z2 ⎞ z ⎛ R ⎞ 3⎛ ⎛R ⎞ 5⎛ ⎛R ⎞ 5⎛ = ⎢ − e3 ⎢1 + J 2 ⎜ e ⎟ ⎜ 1 − 5 2 ⎟ + J 3 ⎜ e ⎟ ⎜ 3 − 7 2 ⎟ − J 4 ⎜ e ⎟ ⎜ 3 − 42 2 + 63 4 ⎟ ⎥ r r ⎠ ⎦⎥ r ⎠ r ⎠r ⎝ r ⎠ 2⎝ ⎝ r ⎠ 2⎝ ⎝ r ⎠ 8⎝ ⎢ r ⎢⎣ ⎢ 2 3 2 4 4 2 2 2 ⎢ − μe z ⎡1 + J ⎛ Re ⎞ 3 ⎛ 3 − 5 z ⎞ + J ⎛ Re ⎞ 5 ⎛ 6 − 7 z ⎞ z ⎤ + μe J ⎛ Re ⎞ 3 − J ⎛ Re ⎞ 5 ⎛ 15 − 70 z + 63 z ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ 2 3 3 4 ⎜ r ⎟ 2 ⎜ r ⎟ 2 ⎜ r ⎟ 8 3 ⎢ r4 r2 ⎠ r 2 ⎠ r ⎥⎦ r 2 ⎜⎝ r ⎟⎠ 2 r2 ⎝ ⎠ ⎝ ⎝ ⎠ ⎝ ⎝ ⎠ ⎝ ⎣ r ⎢⎣

where r = x2 + y 2 + z 2 Re

μe J 2 J3 J 4

distance between the geocentre of the earth and the GEO satellite the earth mean radius (based onWGS-84 reference frame) the gravitational parameter of the earth coefficients are referred to as zonal harmonics, related to the terrestrial model.

The luni-solar effect can be represented as (Montenbruck & Eberhard 2000):

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎞⎥ ⎟⎥ ⎠⎦

⎛ r −r r ⎞ a Sun = μ s ⎜ s − s ⎟ ⎜ r −r 3 r 3 ⎟ s ⎝ s ⎠

⎛ r −r r ⎞ a Moon = μm ⎜ m − m ⎟ ⎜ r −r 3 r 3 ⎟ m ⎝ m ⎠

where μ s and μm are the gravitational parameters of the sun and moon respectively, and T T rs = [ xs , ys , z s ] and rm = [ xm , ym , zm ] are position vectors of the sun and moon in the ECI coordinate frame respectively. The receiver clock is also modelled. The state of the clock part X clock is composed of clock bias and bias rates, i.e. X clock = [δ tu δ tru ] . An exponentially correlated clock model driven by a zero mean white noise is integrated in the process model; referred to as first order Markov processes. It is assumed the four different GNSSs have unified time, hence the clock error is: T

X clock = f clock ( X orbit , t ) + Wclock

⎡ = ⎢δ tu ⎣

T

⎤ 1 − δ tru ⎥ + [ wtu Tru ⎦

wtu ]

T

(7)

The parameter T is the clock model correlation time. The resulting navigation state is a eightstate vector composed of six spacecraft position and velocity components and the receiver clock bias and bias rates. The overall process model can be expressed as: X = f (X ) +W (8) f (X ) X and the measurement noise, in the ECI frame, where the state vector , the state model are: ru

⎡X ⎤ X = ⎢ orbit ⎥ ⎣ X clock ⎦

,

⎡ f ( X , t) ⎤ ⎡W ⎤ f ( X ) = ⎢ orbit orbit ⎥ W = ⎢ orbit ⎥ f t X ( , ) ⎣ clock clock ⎦ , ⎣Wclock ⎦ .

All perturbations W not included in the process model are described as Gaussian errors with E[W ] = 0 and E[WW T ] = Q . The error covariance matrices Q and R are assumed to be diagonal, an important hypothesis using Kalman filter. That means the perturbations and noises in the process and measurement filter models must be uncorrelated. 2.3 The Extended Kalman Filter The blending of GEO satellite state dynamics and pseudo-range measurements has been implemented within a Kalman filter scheme. The typical navigation solution consists of estimates of spacecraft position, velocity and model parameters of the receiver clock behaviour. In this case, the estimation is implemented as an Extended Kalman Filter (EKF) due to the non-linear state dynamics listed above. The main advantages of the Kalman filter approach with respect to other algorithms are: 1) it is a statistical filter that utilises a priori information from the previous epoch in the presence of random perturbations; and 2) the implemented dynamic model makes the filter able to predict position and velocity at a future time, even in the absence of measurements (Campana & Marradi 2000). The filter measurement model and the system dynamic model are equations (2) and(8). In an EKF implementation, the partial derivative matrices of the predicted state vector rate f ( X ) and of the predicted measurement vector h( X ) at the k-th step must be considered:

hk = h( X k )

Hk = H (X k )

fk = f ( X k )

Fk = F ( X k )

∂h( X ) ∂X ∂f ( Xˆ )

H= F=

∂X

where Xˆ and X are the estimated and the predicted state vectors, respectively. From the previous definitions, the correction process consists of the following steps (Minkler & Minkler 1993). The prediction and filtering equations of the EKF consist of the following steps: ˆ (k + 1 / k ) = X ˆ (k / k ) + f [ X(k )] ⋅ T X ˆ (k + 1) = X ˆ (k + 1 / k ) + δ X ˆ (k + 1) X ˆ (k + 1) = K (k + 1){Z(k + 1) − H[ X ˆ (k + 1/ k )]} δX P (k + 1 / k ) = Φ(k + 1 / k )P (k / k )Φ(k + 1 / k )T + Q K (k + 1) = P(k + 1 / k )H (k + 1)T [H(k + 1) P(k + 1 / k ) H(k + 1)T + R (k + 1)]−1 P (k + 1) = [I − K (k + 1)H (k + 1)]P (k + 1 / k )[I − K (k + 1)H (k + 1)]T + K (k + 1)R (k + 1)K (k + 1)T

where P is the covariance matrix of the state estimation uncertainty, K is the Kalman gain matrix, Φ is the state transition matrix of the discrete dynamic system Φ ≈ I + F ⋅ T , and T is the integration interval of the filter. 3 SIMULATION AND ORBIT DETERMINATION STRATEGY 3.1 Simulation Description A theoretical approach to the GEO determination has been described in the previous section. Now the proposed algorithm is implemented and its performance is analysed using a MonteCarlo simulation (Minkler & Minkler 1993). The test scenario given in Figure 2 is used to analyse the proposed algorithm which includes the following modules: GNSS satellite database, GEO satellite orbit dynamics, computation of simulated measurements and Kalman filter process. GNSS Satellite Database

GNSS Positon Computation

User Satellite Orbit Dynamics

Access Computation

rGNSS

Transition Marix and State Noise Marix Computation

φ, Q State Vector and Covariance Propagation

Measrement Processing and Observation Matrix Computation

P, x

H, z Kalman Filter Update

Figure 2: Orbit determination algorithm architecture. The Satellite Tool Kit (STK, v5.0) software (www.stk.com) is used to produce satellite ephemeris to establish the GNSS Satellite Database. The initial almanac (i.e. osculating orbit

parameters) of GPS and GLONASS constellations was adopted from the web sources (Table 1). Constellation GPS GLONASS GALILEO COMPASS

Sat Number 31 24 30 35

Almanac Source http://www.glonass-ianc.rsa.ru http://www.glonass-ianc.rsa.ru http://www.esa.int/esaNA/galileo.html (Tan 2008)) Table 1. Almanac sources

In the simulation, the antenna half-cone is set as follows: • Navigation satellite transmit antenna half-cone angle: 21.3° (Moreau et al. 2001)（Only consider receiving the main lobe） • User satellite receiver antenna half-cone angle: 90°（main lobe） The above tow angle is considered only for the main lobe angle. The test simulations for the algorithms have been conducted for a 24-hour dataset. The test epoch starts on 30 April 2009 at 12:00:00.00 UTCG, and stops on 1 May 2009 at 12:00:00 UTCG. 3.2 GEO Tracking Performance

5

5

4

4

GPS

GPS

Initial tests were conducted for several GEO scenarios to assess the feasibility of tracking GNSS satellites. The scenario assumes the GNSS receiver has a down-looking antenna to the earth. Here the impact of the weak signal tracking capabilities is not discussed. Satellites are considered as visible and available if their line-of-sight is inside both the transmission cone and the reception cone, and does not intersect the earth. This section will present the results from the tests conducted for four geostationary orbits, where the longitude of the four satellites are 0, 90, 180 and 270 degrees. 3 2 1 0

3 2 1

0

1

2

3

4

5

6

7

8

0

9

0

1

2

3

4

5

6

7

8

4

x 10 5

GLONASS

GLONASS

5 4 3 2 1 0

0

1

2

3

4

5

6

7

8

4 3 2 1 0

9

0

1

2

3

4

5

6

7

8

4

x 10 5

GALILEO

GALILEO

5 4 3 2 1 0

1

2

3

4

5

6

7

8

4 3 2 1 0

9

0

1

2

3

4

5

6

7

8

4

x 10

COMPASS

COMPASS

5 4 3 2 1 0

1

2

3

4

5

Time(s)

0°GEO

6

7

8

9 4

x 10

9 4

x 10

0

9 4

x 10

0

9 4

x 10

5 4 3 2 1 0

0

1

2

3

4

5

Time(s)

90°GEO

6

7

8

9 4

x 10

5

4

4

GPS

GPS

5

3 2 1 0

3 2 1

0

1

2

3

4

5

6

7

8

0

9

0

1

2

3

4

5

6

7

8

9

4

4

x 10

x 10 5

GLONASS

GLONASS

5 4 3 2 1 0

0

1

2

3

4

5

6

7

8

4 3 2 1 0

9

0

1

2

3

4

5

6

7

8

9

4

4

x 10

x 10 5

GALILEO

GALILEO

5 4 3 2 1 0

0

1

2

3

4

5

6

7

8

4 3 2 1 0

9

0

1

2

3

4

5

6

7

8

9

4

4

x 10

5

COMPASS

COMPASS

x 10

4 3 2 1 0

0

1

2

3

4

5

6

7

8

5 4 3 2 1 0

9

0

1

2

3

4

4

Time(s)

5

6

7

8

9 4

Time(s)

x 10

180°GEO

x 10

270°GEO

Figure 3: Comparison of single GNSS satellites in view versus multi-GNSS system for four GEO satellite configurations.

11

11

10

10

Number of Visible Satellite

Number of Visible Satellite

Figure 3 shows that there are zero satellites tracked using only a single GNSS. Because the number of visible satellites using a single system is less than four, classical least squares single-epoch solutions are not possible. The number of available satellites increases using multi-GNSS system, and there are no satellite gaps.

9 8 7 6 5 4 3 2 1 0 0

9 8 7 6 5 4 3 2 1

1

2

3

4

5

6

7

8

Time(s)

0 0

9

1

2

3

5

6

7

8

9 4

x 10

90°GEO

11

11

10

10

9

9

Number of Visible Satellite

Number of Visible Satellite

0°GEO

8 7 6 5 4 3 2 1 0 0

4

Time(s)

4

x 10

8 7 6 5 4 3 2 1

1

2

3

4

5

Time(s)

180°GEO

6

7

8

9 4

x 10

0 0

1

2

3

4

5

6

7

Time(s)

270°GEO

Figure 4: Comparison of the multi-GNSS visibility conditions for four GEO satellite configurations.

8

9 4

x 10

Figure 4 shows a comparison of the number of satellites tracked in the four different GEO satellite configurations. The availability is different depending on the longitude. From these four plots it can be seen that the receiver always tracked at least one satellite, and there are many instances when there were four or more satellites being tracked. The percentage of four or more satellites that were tracked continuously are shown in Table 2. It indicates that in 90° GEO case, the GEO satellite can receive four or more navigation signals continuously, the longest time occupied being about 11% of the whole running time. This continuity ensures the convergence of the filter. Continuity 0°GEO 90°GEO 180°GEO 270°GEO

Minimum (%) 3.264 0.069 0.069 4.861

Longest(%) 36.806 11.042 11.806 19.653

20

20

18

18

16

16

14

14

Percentage

Percentage

Table 2: Percentage of four or more satellites continuously tracking time.

12 10 8

12 10 8

6

6

4

4

2

2

0

0

1

2

3

4

5

6

7

8

9

0

10

0

Number of Multi GNSS Visible Satellite

1

2

0°GEO 18

18

16

16

5

6

7

8

9

10

14

14

12

Percentage

Percentage

4

90°GEO

20

12 10 8

10 8 6

6

4

4

2

2 0

3

Number of Multi GNSS Visible Satellite

0

1

2

3

4

5

6

7

8

9

Number of Multi GNSS Visible Satellite

180°GEO

10

0

0

1

2

3

4

5

6

7

8

9

10

Number of Multi GNSS Visible Satellite

270°GEO

Figure 5: Comparison of the percentage of N satellites tracked for four GEO satellite configurations. Figure 5 shows histograms indicating the percentage of time one, two, three, etc., satellites tracked. The statistical data is given in Table 3. Figure 5 and Table 3 indicate that less than 4 satellites were tracked less than 30% of the time, four or more satellites were tracked more than 70% of the time which also ensures the convergence of the filter.

Number of Satellite 0 1 2 3 4 5 6 7 8 9 10

0°GEO

90°GEO

180°GEO

270°GEO

0 0 0.694 2.500 8.750 18.611 15.764 12.222 18.750 16.319 6.389

0.278 4.375 11.736 13.958 10.069 13.472 18.056 15.069 9.583 1.458 1.944

0.278 1.875 11.042 17.639 19.028 10.416 15.347 13.264 7.847 2.222 1.042

0 0 0 2.708 10.208 17.778 16.944 16.528 12.778 14.931 8.125

Table 3: Percentage of n satellites were tracked. 2.4 Filtering Results Choosing the 90° GEO configuration to implement the Kalman filter, its elements are presented in Table 4 Compared to any single GNSS, the number and the duration are improved due to the multiGNSSs adopted. Therefore it is more suitable for using a navigation filter for orbit determination in multi-GNSSs scenario. Orbit type Cartesian (J2000) X(m) -26,214,220,335.009 Y(m) 33,024,715,796.160 Z(m) 0 Vx(m/s) -2,408.201 Vy(m/s) -1,911.571 Vz(m/s) 0.000

Geostationary Keplerian (J2000) Semi-major Axis (m) 42,164,169.637 Eccentricity 1.42579e-15 Inclination (°) 0 RA Ascn. Node (°) 0 Long. Ascn. Node (°) 90 True Anomaly (°) 128.441696

Table 4: Orbital elements for 90° GEO configuration orbits. The covariance matrix of process noise

W(k )

is assumed as:

Q(k ) = diag ([(0.02m) , (0.02m) , (0.02m) , (2 × 10 m / s) 2 , (2 × 10-4 m / s) 2 , (2 ×10-4 m / s) 2 , (0.5m) 2 , (1× 10-6 m) 2 ]) 2

2

2

-4

The initial errors of the satellite position and velocity are: δ X(0) = [1km, 1km, 1km, 10m/s, 10m/s, 10m/s,300m,0.0003m ] The initial covariance matrix of the satellite position and velocity is: Pg (0)=diag ([(1km)2 , (1km)2 , (1km)2 ,( 10m / s) 2 , ( 10m / s) 2 , ( 10m / s) 2 , ( 300m) 2 , ( 0.0003m) 2 ])

The errors of position and velocity are assumed to be: δ r = ( xˆ − x) 2 + ( yˆ − y ) 2 + ( zˆ − z ) 2 δ v = (vˆx − vx ) 2 + (vˆ y − v y ) 2 + (vˆz − vz ) 2

where x, y, z, v , v , v are the true values of the satellite position and velocity, and xˆ, yˆ , zˆ, vˆ , vˆ , vˆ are the estimated values. If the value of the position error δ r is recorded for N different samples (δ r , δ r ,…δ r ) , then the root mean square (RMS) of δ r is defined as: x

1

2

y

z

x

y

z

N

1 N ∑ δ ri2 N i =1

RMS (δ r ) = E (δ r 2 ) ≅

The RMS of

δv

can also be defined as: 1 N ∑ δ vi2 N i =1

RMS (δ v) = E (δ v 2 ) ≅

Two settings were tested: • Using four or less visible navigation satellite signals. • Using all the visible navigation satellite signals.

0.5

1000 Position Error Estimation Standard Deviation

900

0.4 Velocity Error Estimation(m/s)

800

Position Error(m)

700 600 500 400 300

0.35 0.3 0.25 0.2 0.15 0.1

200

0.05

100 0

Velocity Error Estimation Standard Deviation

0.45

0 0

1

2

3

4 5 Time(s)

6

7

8

9

0

1

2

3

4 5 Time(s)

4

6

7

8

9 4

x 10

x 10

Figure 6: Convergence of Kalman filter (four or less the visible navigation satellites). 0.5

1000 Position Error Estimation Standard Deviation

900

0.4 Velocity Error Estimation(m/s)

800

Position Error(m)

700 600 500 400 300

0.35 0.3 0.25 0.2 0.15 0.1

200

0.05

100 0 0

Velocity Error Estimation Standard Deviation

0.45

1

2

3

4

5 Time(s)

6

7

8

9 4

0 0

1

2

3

4

5

6

Time(s)

x 10

7

8

9 4

x 10

Figure 7: Convergence of Kalman filter (all visible navigation satellites). Compare the test results between both settings. The P matrix is convergent during the elapsed hours, which represents the covariance of the estimate errors (Qin et al. 1998). The rebound in the lower branch of the orbit is due to the loss of orbit dynamics propagation accuracy. However in the second setting test, the convergence is quicker. From the plot (see Figure 6), the RMS errors of the satellite position and velocity of the integrated navigation solution are 185.776m (1σ) and 0.073m/s (1σ) for the GEO orbit. In Figure 7, an accuracy of 157.767m (1σ) and 0.096m/s (1σ) is reached.

3. CONCLUDING REMARKS The exploitation of multi-GNSS signals for precise orbit determination for GEO satellites is not only possible but also practical from the simulation study presented in this paper. No single GNSS can solve the problem of GEO satellites' line of sight non-availability, but multiGNSS improves the navigation performance. For orbit determination the Extend Kalman filter is a well-known technique, but it requires accurate orbit dynamics modelling. Using multi-GNSS with an orbit determination filter is a viable option for GEO missions to sequentially process sparsely available pseudo-range measurements. In this paper the navigation performance using the extended Kalman filter with the hypothesis that all the four GNSSs are available was investigated. Even when using the main lobe only, the accuracy remains within an acceptable range (< 200m 1σ for position vectors and < 0.1m/s 1σ for velocity vectors), therefore it is concluded that this algorithm meets the requirement for a GEO satellite’s autonomous navigation. Although the algorithm is designed for a GEO satellite and the test results are based on simulated measurements, it is expected to be also applicable to HEO satellites as well. Even so, this algorithm still requires further investigation and testing when multi-GNSS signals become available. ACKNOWLEDGMENTS The authors wish to thank their colleagues from the School of Surveying and Spatial Information Systems, University of New South Wales, and the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics. The first author wishes to thank the China Scholarship Council for the scholarship support during her joint training. This paper is extracted and modified from a paper that was presented at the Institute of Navigation (ION) GNSS conference, Savannah, Georgia, USA, 22-25 September 2009. REFERENCES Bauer F., Hartman K., Lightsey E., Center N. and Greenbelt M. (1998). Spaceborne GPS Current Status and Future Visions. Proceedings of ION GPS-98, 1493-1508. Campana R. and Marradi L. (2000). GPS-based Space Navigation: Comparison of Kalman Filtering Schemes. Proceedings of ION GPS 2000, Salt Lake City, Utah, 1636-1645. Ebinuma T. and Unwin M. (2007). GPS Receiver Demonstration on a Galileo Test Bed Satellite. Journal of Navigation 60(03): 349-362. Eissfeller B., Balbach O., Rossbach U. and Fraile-Ordonez J. (1996). GPS Navigation on the HEO Satellite Mission EQUATOR-S - Results of the Feasibility Study. Proceedings of ION GPS-96, Kansas City, Missouri, Institute of Navigation, 1331-1340. Krauss P. A., Kühl C., Mittnacht M., Heim J. and Gottzein E. (2008). Modernized Spaceborne GNSS Receivers. Proceedings of Proceedings of the 17th World Congress The International Federation of Automatic Control, Seoul, South Korea, 4707-4712. Kronman J. (2000). Experience Using GPS for Orbit Determination of a Geosynchronous Satellite. Proceedings of ION GPS 2000, Salt Lake City, Utah, 1622-1626.

Martin-Mur T., Dow J. and Bondarenco N. (1995). Use of GPS for Precise and Operational Orbit Determination at ESOC. Proceedings of ION GPS-95, Palm Springs, California, 619-626. Mehlen C. and Laurichesse D. (2001). Real-time GEO Orbit Determination Using TOPSTAR 3000 GPS Receiver. Navigation 48(3): 169-179. Melbourne W., Yunck T., Bertiger W., Haines B. and Davis E. (1994). Scientific Applications of GPS on Low Earth Orbiters. Stuttgart, Germany, JPL Technical Report Server, DSpace at Jet Propulsion Laboratory, Pasadena, United States. Minkler G. and Minkler J. (1993). Theory and Application of Kalman Filtering, Magellan Book Company, Palm Bay, Florida, USA. Mittnacht M. and Fichter W. (2000). Real Time On-board Orbit Determination of GEO Satellites Using Software or Hardware Correlation. Proceedings of ION GPS 2000, Salt Lake City, Utah, 1976-1984. Montenbruck O. and Eberhard G. (2000). Satellite Orbits: Models, Methods, and Applications. Berlin, Springer Verlag. Moreau M.C., Axelrad P., Wennersten M. and Long A.C. (2001). Test Results of the PiVoT Receiver in High Earth Orbits using a GSS GPS Simulator. Proceedings of ION GPS 2001, Salt Lake City, Utah, 2316-2326. Munjal P., Feess W. and Ananda M. (1992). A Review of Spaceborne Application of GPS. Proceedings of ION GPS-92, Albuquerque, New Mexico, 813-823. Qin Y., Zhang H. and Wang S. (1998). Kalman Filtering and Principle of Integrated Navigation. Xian, China, Northwestern Polytechnical University Press. Tan S. (2008). Development and Though of Compass Navigation Satellite System. Journal of Astronautics 29(2): 391-396. Vallado D.A. and McClain W.D. (2001). Fundamentals of Astrodynamics and Applications. Boston MA, Space Technology Library, Kluwer Academic Publishers. Vasilyev M. (1999). Real Time Autonomous Orbit Determination of GEO Satellite Using GPS. Proceedings of ION GPS-99, Nashiville, Tennesee, 451-457.